A modification of Galerkin's method for option pricing

Mikhail Dokuchaev; Guanglu Zhou; Song Wang

doi:10.3934/jimo.2021077

Article Contents

2022, Volume 18, Issue 4: 2483-2504. Doi: 10.3934/jimo.2021077

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A modification of Galerkin's method for option pricing

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, Australia

^* Corresponding author: Mikhail Dokuchaev
^* Corresponding author: Mikhail Dokuchaev

Received: June 2020

Revised: January 2021

Published: July 2022

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Abstract

We present a novel method for solving a complicated form of a partial differential equation called the Black-Scholes equation arising from pricing European options. The novelty of this method is that we consider two terms of the equation, namely the volatility and dividend, as variables dependent on the state price. We develop a Galerkin finite element method to solve the problem. More specifically, we discretize the system along the state variable and build new basis functions which we use to approximate the solution. We establish convergence of the proposed method and numerical results are reported to show the proposed method is accurate and efficient.

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Mathematics Subject Classification: Primary: 15A18, 90C30, 90C33.

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Figure 1. Basis function $ \phi_k(y) $ for $ y_{k-1} = -1 $, $ y_k = 0 $, $ y_{k+1} = 1 $, $ \rho = 0.045 $, and $ \eta = -0.045 $

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Figure 2. Comparison of exact solution U and numerical solution V

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Figure 3. Comparison of exact function U and numerical solution V for the case of non-constant $ \sigma $

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Table 1. Error of calculation of the put option for r = 0

$ N $, $ N_t $	E
20, 20	0.003902114
40, 40	0.006529201
80, 80	0.007018533
160,160	0.003593509
320,320	0.0003050627
640,640	7.043937e-05

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Table 2. Error of calculation of the put option for r = 0.025

$ N $, $ N_t $	E
20, 20	0.003319792
40, 40	0.005971542
80, 80	0.006657621
160,160	0.003484265
320,320	0.001151376
640,640	0.0003031325

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Table 3. Error of calculation of the put option for r = 0.05

$ N $, $ N_t $	E
20, 20	0.002830843
40, 40	0.00547276
80, 80	0.006323301
160,160	0.003382576
320,320	0.001133573
640,640	0.0003015444

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Table 4. Error of calculation of the case of state-dependent volatility

$ N $, $ N_t $	E
20, 20	8.60202
40, 40	0.1838133
80, 80	0.09596427
160,160	0.04898591
320,320	0.02474154
640,640	0.01243255

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